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        <p>![](<a href="https://gitee.com/wang_wx/image_bed/raw/master/201910/2019-10-01" target="_blank" rel="noopener">https://gitee.com/wang_wx/image_bed/raw/master/201910/2019-10-01</a> 09.35.09 1112345.jpg) </p>
<center><font size=1.5>20190928-陕西咸阳旬邑 </center>



<p>数学上，<strong>克罗内克积</strong>（Kronecker product）是两个任意大小的矩阵间的运算，表示为⊗ 。克罗内克积是<a href="https://www.wikiwand.com/zh-cn/外积" target="_blank" rel="noopener">外积</a>从向量到矩阵的推广，也是<a href="https://www.wikiwand.com/zh-cn/张量积" target="_blank" rel="noopener">张量积</a>在标准基下的矩阵表示。 本文来自<a href="https://www.wikiwand.com/zh-cn/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF" target="_blank" rel="noopener">维基百科</a>（需科学上网）。</p>
<a id="more"></a>

<blockquote>
<p><a href="https://www.wikiwand.com/zh-cn/外积" target="_blank" rel="noopener">外积</a></p>
<p>外积（英语：Outer product），在线性代数中一般指两个向量的张量积，其结果为一矩阵；与外积相对，两向量的内积结果为标量。 外积也可视作是矩阵的克罗内克积的一种特例。</p>
<p><a href="https://www.wikiwand.com/zh-cn/%E5%BC%A0%E9%87%8F%E7%A7%AF" target="_blank" rel="noopener">张量积</a></p>
<p>在数学中，张量积，记为  ⊗ ，可以应用于不同的上下文中,如向量、矩阵、张量、向量空间、代数、拓扑向量空间和模。在各种情况下这个符号的意义是同样的:最一般的双线性运算。</p>
</blockquote>
<p>尽管没有明显证据证明德国数学家<a href="https://www.wikiwand.com/zh-cn/利奥波德·克罗内克" target="_blank" rel="noopener">利奥波德·克罗内克</a>是第一个定义并使用这一运算的人，<strong>克罗内克积</strong>还是以其名字命名。确实，在历史上，<strong>克罗内克积</strong>曾以Johann Georg Zehfuss名字命名为Zehfuss矩阵。 </p>
<h2 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h2><p>如果<em>A</em>是一个 <em>m</em> × <em>n</em> 的矩阵，而<em>B</em>是一个 <em>p</em> × <em>q</em> 的矩阵，<strong>克罗内克积</strong>$A \otimes B$ 则是一个 <em>mp</em> × <em>nq</em> 的<a href="https://www.wikiwand.com/zh-cn/分塊矩陣" target="_blank" rel="noopener">分块矩阵</a><br>$$<br>A \otimes B=\left[\begin{array}{ccc}{a_{11} B} &amp; {\cdots} &amp; {a_{1 n} B} \ {\vdots} &amp; {\ddots} &amp; {\vdots} \ {a_{m 1} B} &amp; {\cdots} &amp; {a_{m n} B}\end{array}\right]<br>$$</p>
<blockquote>
<p><a href="https://www.wikiwand.com/zh-cn/分塊矩陣" target="_blank" rel="noopener">分块矩阵</a> </p>
<p>在数学的矩阵理论中，一个分块矩阵或是分段矩阵就是将矩阵分割出较小的矩形矩阵，这些较小的矩阵就称为区块。换个方式来说，就是以较小的矩阵组合成一个矩阵。</p>
</blockquote>
<p>更具体地可表示为 </p>
<p><img src="https://cdn.mathpix.com/snip/images/4PnYMkK1lPNJG3p4SwsEjfLPID2s_R0uNEmWts2Y1Bc.original.fullsize.png" alt=""></p>
<p>我们可以更紧凑地写为 <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5740426c7e1381c12587ec57d22da1c52e0f4968" alt="{\displaystyle (A\otimes B)_{p(r-1)+v,q(s-1)+w}=a_{rs}b_{vw))"> </p>
<h3 id="例子"><a href="#例子" class="headerlink" title="例子"></a>例子</h3><p>$$<br>\left[\begin{array}{ll}{1} &amp; {2} \ {3} &amp; {1}\end{array}\right] \otimes\left[\begin{array}{cc}{0} &amp; {3} \ {2} &amp; {1}\end{array}\right]=\left[\begin{array}{llll}{1 \cdot 0} &amp; {1 \cdot 3} &amp; {2 \cdot 0} &amp; {2 \cdot 3} \ {1 \cdot 2} &amp; {1 \cdot 1} &amp; {2 \cdot 2} &amp; {2 \cdot 1} \ {3 \cdot 0} &amp; {3 \cdot 3} &amp; {1 \cdot 0} &amp; {1 \cdot 3} \ {3 \cdot 2} &amp; {3 \cdot 1} &amp; {1 \cdot 2} &amp; {1 \cdot 1}\end{array}\right]=\left[\begin{array}{llll}{0} &amp; {3} &amp; {0} &amp; {6} \ {2} &amp; {1} &amp; {4} &amp; {2} \ {0} &amp; {9} &amp; {0} &amp; {3} \ {6} &amp; {3} &amp; {2} &amp; {1}\end{array}\right]<br>$$</p>
<p>$$<br>\left[\begin{array}{ll}{1} &amp; {2} \ {3} &amp; {4}\end{array}\right] \otimes\left[\begin{array}{ll}{0} &amp; {5} \ {6} &amp; {7}\end{array}\right]=\left[\begin{array}{lll}{1} &amp; {0} &amp; {5} \ {6} &amp; {7} &amp; {2} \ {3} &amp; {\left[\begin{array}{ll}{0} &amp; {5} \ {6} &amp; {7}\end{array}\right]} &amp; {4\left[\begin{array}{ll}{0} &amp; {5} \ {6} &amp; {7}\end{array}\right]}\end{array}\right]=\left[\begin{array}{lllll}{1 \times 0} &amp; {1 \times 5} &amp; {2 \times 0} &amp; {2 \times 5} \ {1 \times 6} &amp; {1 \times 7} &amp; {2 \times 6} &amp; {2 \times 7} \ {3 \times 0} &amp; {3 \times 5} &amp; {4 \times 0} &amp; {4 \times 5} \ {3 \times 6} &amp; {3 \times 7} &amp; {4 \times 6} &amp; {4 \times 7}\end{array}\right]=\left[\begin{array}{cccc}{0} &amp; {5} &amp; {0} &amp; {10} \ {6} &amp; {7} &amp; {12} &amp; {14} \ {0} &amp; {15} &amp; {0} &amp; {20} \ {18} &amp; {21} &amp; {24} &amp; {28}\end{array}\right]<br>$$</p>
<h2 id="特性"><a href="#特性" class="headerlink" title="特性"></a>特性</h2><h3 id="双线性和结合律"><a href="#双线性和结合律" class="headerlink" title="双线性和结合律"></a>双线性和结合律</h3><p><strong>克罗内克积</strong>是<a href="https://www.wikiwand.com/zh-cn/张量积" target="_blank" rel="noopener">张量积</a>的特殊形式，因此满足<a href="https://www.wikiwand.com/zh-cn/双线性映射" target="_blank" rel="noopener">双线性</a>与<a href="https://www.wikiwand.com/zh-cn/结合律" target="_blank" rel="noopener">结合律</a>：<br>$$<br>\begin{array}{l}{A \otimes(B+C)=A \otimes B+A \otimes C \quad \text { (if } B \text { and } C \text { have the same size) }} \ {(A+B) \otimes C=A \otimes C+B \otimes C \quad \text { (if } A \text { and } B \text { have the same size) }} \ {(k A) \otimes B=A \otimes(k B)=k(A \otimes B)} \ {(A \otimes B) \otimes C=A \otimes(B \otimes C)}\end{array}<br>$$<br>其中，<em>A</em>, <em>B</em> 和 <em>C</em> 是矩阵，而 <em>k</em> 是常量。 </p>
<p><strong>克罗内克积</strong>不符合<a href="https://www.wikiwand.com/zh-cn/交换律" target="_blank" rel="noopener">交换律</a>：通常，$A \otimes B$ 不同于 $B \otimes A$ 。  </p>
<p>$A \otimes B$ 和是$B \otimes A$排列等价的，也就是说，存在<a href="https://www.wikiwand.com/zh-cn/排列矩陣" target="_blank" rel="noopener">排列矩阵</a><em>P</em>和<em>Q</em>，使得<br>$$<br>A \otimes B=P(B \otimes A) Q<br>$$<br>如果<em>A</em>和<em>B</em>是方块矩阵，则$A \otimes B$ 和$B \otimes A$甚至是排列<a href="https://www.wikiwand.com/zh-cn/相似矩陣" target="_blank" rel="noopener">相似</a>的，也就是说，我们可以取<em>P</em> = <em>Q</em>T。 </p>
<h3 id="混合乘积性质"><a href="#混合乘积性质" class="headerlink" title="混合乘积性质"></a>混合乘积性质</h3><p>如果<strong>A</strong>、<strong>B</strong>、<strong>C</strong>和<strong>D</strong>是四个矩阵，且矩阵乘积<strong>AC</strong>和<strong>BD</strong>存在，那么：<br>$$<br>(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D})=\mathbf{A} \mathbf{C} \otimes \mathbf{B D}<br>$$<br>这个性质称为“混合乘积性质”，因为它混合了通常的矩阵乘积和克罗内克积。于是可以推出，$A \otimes B$ 是<a href="https://www.wikiwand.com/zh-cn/可逆矩阵" target="_blank" rel="noopener">可逆</a>的<a href="https://www.wikiwand.com/zh-cn/当且仅当" target="_blank" rel="noopener">当且仅当</a><strong>A</strong>和<strong>B</strong>是可逆的，其逆矩阵为：<br>$$<br>(\mathbf{A} \otimes \mathbf{B})^{-1}=\mathbf{A}^{-1} \otimes \mathbf{B}^{-1}<br>$$</p>
<h3 id="克罗内克和"><a href="#克罗内克和" class="headerlink" title="克罗内克和"></a>克罗内克和</h3><p>如果<strong>A</strong>是<em>n</em> × <em>n</em>矩阵，<strong>B</strong>是<em>m</em> × <em>m</em>矩阵，$\mathbf{I}<em>{k}$表示<em>k</em> × <em>k</em>单位矩阵，那么我们可以定义克罗内克和$\otimes $为：<br>$$<br>\mathbf{A} \oplus \mathbf{B}=\mathbf{A} \otimes \mathbf{I}</em>{m}+\mathbf{I}_{n} \otimes \mathbf{B}<br>$$</p>
<h3 id="谱"><a href="#谱" class="headerlink" title="谱"></a>谱</h3><p>假设<strong>A</strong>和<strong>B</strong>分别是大小为<em>n</em>和<em>q</em>的方块矩阵。设λ1，……，λ<em>n</em>为<strong>A</strong>的<a href="https://www.wikiwand.com/zh-cn/特征值" target="_blank" rel="noopener">特征值</a>，μ1，……，μ<em>q</em>为<strong>B</strong>的特征值。那么$A \otimes B$的特征值为：<br>$$<br>\lambda_{i} \mu_{j}, \quad i=1, \ldots, n, j=1, \ldots, q<br>$$<br>于是可以推出，两个矩阵的克罗内克积的<a href="https://www.wikiwand.com/zh-cn/迹" target="_blank" rel="noopener">迹</a>和<a href="https://www.wikiwand.com/zh-cn/行列式" target="_blank" rel="noopener">行列式</a>为：<br>$$<br>\operatorname{tr}(\mathbf{A} \otimes \mathbf{B})=\operatorname{tr} \mathbf{A} \operatorname{tr} \mathbf{B} \quad \text { and } \quad \operatorname{det}(\mathbf{A} \otimes \mathbf{B})=(\operatorname{det} \mathbf{A})^{q}(\operatorname{det} \mathbf{B})^{n}<br>$$</p>
<h3 id="奇异值"><a href="#奇异值" class="headerlink" title="奇异值"></a>奇异值</h3><p>如果<strong>A</strong>和<strong>B</strong>是长方矩阵，那么我们可以考虑它们的<a href="https://www.wikiwand.com/zh-cn/奇异值分解" target="_blank" rel="noopener">奇异值</a>。假设<strong>A</strong>有<em>r**</em>A<strong>个非零的奇异值，它们是：<br>$$<br>\sigma_{\mathbf{A}, i}, \quad i=1, \dots, r_{\mathbf{A}}<br>$$<br>类似地，设</strong>B**的非零奇异值为：<br>$$<br>\sigma_{\mathbf{B}, i}, \quad i=1, \dots, r_{\mathbf{B}}<br>$$<br>那么克罗内克积$A \otimes B$有个$r_{\mathbf{A}} r_{\mathbf{B}}$非零奇异值，它们是：<br>$$<br>\sigma_{\mathbf{A}, i} \sigma_{\mathbf{B}, j}, \quad i=1, \ldots, r_{\mathbf{A}}, j=1, \ldots, r_{\mathbf{B}}<br>$$<br>由于一个<a href="https://www.wikiwand.com/zh-cn/矩阵的秩" target="_blank" rel="noopener">矩阵的秩</a>等于非零奇异值的数目，因此我们有：<br>$$<br>\operatorname{rank}(\mathbf{A} \otimes \mathbf{B})=\operatorname{rank} \mathbf{A} \operatorname{rank} \mathbf{B}<br>$$</p>
<h3 id="与抽象张量积的关系"><a href="#与抽象张量积的关系" class="headerlink" title="与抽象张量积的关系"></a>与抽象张量积的关系</h3><p>矩阵的克罗内克积对应于线性映射的抽象张量积。特别地，如果向量空间<em>V</em>、<em>W</em>、<em>X</em>和<em>Y</em>分别具有基{v1, … , vm}、 {w1, … , wn}、{x1, … , xd}和{y1, … , ye}，且矩阵<em>A</em>和<em>B</em>分别在恰当的基中表示线性变换<em>S</em> : <em>V</em> → <em>X</em>和<em>T</em> : <em>W</em> → <em>Y</em>，那么矩阵<em>A</em> ⊗ <em>B</em>表示两个映射的张量积<em>S</em> ⊗ <em>T</em> : <em>V</em> ⊗ <em>W</em> → <em>X</em> ⊗ <em>Y</em>，关于<em>V</em> ⊗ <em>W</em>的基{v1 ⊗ w1, v1 ⊗ w2, … , v2 ⊗ w1, … , vm ⊗ wn}和<em>X</em> ⊗ <em>Y</em>的类似基。<a href="https://www.wikiwand.com/zh-cn/克罗内克积#citenote1" target="_blank" rel="noopener">[1]</a> </p>
<h3 id="与图的乘积的关系"><a href="#与图的乘积的关系" class="headerlink" title="与图的乘积的关系"></a>与图的乘积的关系</h3><p>两个<a href="https://www.wikiwand.com/zh-cn/图_(数学)" target="_blank" rel="noopener">图</a>的<a href="https://www.wikiwand.com/zh-cn/邻接矩阵" target="_blank" rel="noopener">邻接矩阵</a>的克罗内克积是它们的张量积图的邻接矩阵。两个图的邻接矩阵的克罗内克和，则是它们的笛卡儿积图的邻接矩阵。参见<a href="https://www.wikiwand.com/zh-cn/克罗内克积#citenoteTAOCP0a2" target="_blank" rel="noopener">[2]</a>第96个练习的答案。 </p>
<h3 id="转置"><a href="#转置" class="headerlink" title="转置"></a>转置</h3><p><strong>克罗内克积</strong>转置运算符合分配律：<br>$$<br>(A \otimes B)^{T}=A^{T} \otimes B^{T}<br>$$</p>
<h2 id="矩阵方程"><a href="#矩阵方程" class="headerlink" title="矩阵方程"></a>矩阵方程</h2><p>克罗内克积可以用来为一些矩阵方程得出方便的表示法。例如，考虑方程<em>AXB</em> = <em>C</em>，其中<em>A</em>、<em>B</em>和<em>C</em>是给定的矩阵，<em>X</em>是未知的矩阵。我们可以把这个方程重写为 </p>
<p>$$<br>\left(B^{T} \otimes A\right) \operatorname{vec}(X)=\operatorname{vec}(A X B)=\operatorname{vec}(C)<br>$$<br>这样，从克罗内克积的性质可以推出，方程<em>AXB</em> = <em>C</em>具有唯一的解，当且仅当<em>A</em>和<em>B</em>是非奇异矩阵。（<a href="https://www.wikiwand.com/zh-cn/克罗内克积#CITEREFHornJohnson1991" target="_blank" rel="noopener">Horn &amp; Johnson 1991</a>，Lemma 4.3.1）. </p>
<p>在这里，vec(<em>X</em>)表示矩阵<em>X</em>的向量化，它是把<em>X</em>的所有列堆起来所形成的<a href="https://www.wikiwand.com/zh-cn/列向量" target="_blank" rel="noopener">列向量</a>。 </p>
<p>如果把<em>X</em>的行堆起来，形成列向量<em>x</em>，则$A X B$也可以写为$\left(A \otimes B^{T}\right) x$（<a href="https://www.wikiwand.com/zh-cn/克罗内克积#CITEREFJain1989" target="_blank" rel="noopener">Jain 1989</a>，2.8 block Matrices and Kronecker Products)。 </p>
<h2 id="参考文献"><a href="#参考文献" class="headerlink" title="参考文献"></a>参考文献</h2><ol>
<li><strong>^</strong> Pages 401–402 of Dummit, David S.; Foote, Richard M., Abstract Algebra 2, New York: John Wiley and Sons, Inc., 1999, <a href="https://www.wikiwand.com/zh-cn/Special:网络书源/0-471-36857-1" target="_blank" rel="noopener">ISBN 0-471-36857-1</a> </li>
<li><strong>^</strong> D. E. Knuth:  <a href="http://www-cs-faculty.stanford.edu/~knuth/fasc0a.ps.gz" target="_blank" rel="noopener">“Pre-Fascicle 0a: Introduction to Combinatorial Algorithms”</a>, zeroth printing (revision 2), to appear as part of D.E. Knuth: <em>The Art of Computer Programming Vol. 4A</em> </li>
</ol>
<ul>
<li>Horn, Roger A.; Johnson, Charles R., Topics in Matrix Analysis, Cambridge University Press, 1991, <a href="https://www.wikiwand.com/zh-cn/Special:网络书源/0-521-46713-6" target="_blank" rel="noopener">ISBN 0-521-46713-6</a>.</li>
<li>Jain, Anil K., Fundamentals of Digital Image Processing, Prentice Hall, 1989, <a href="https://www.wikiwand.com/zh-cn/Special:网络书源/0-13-336165-9" target="_blank" rel="noopener">ISBN 0-13-336165-9</a>.</li>
<li><a href="https://www.wikiwand.com/zh-cn/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF" target="_blank" rel="noopener">https://www.wikiwand.com/zh-cn/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF</a></li>
<li><a href="https://www.wikiwand.com/en/Kronecker_product" target="_blank" rel="noopener">https://www.wikiwand.com/en/Kronecker_product</a></li>
<li><a href="https://www.wikiwand.com/en/Roger_Horn" target="_blank" rel="noopener">Horn, Roger A.</a>; <a href="https://www.wikiwand.com/en/Charles_R._Johnson" target="_blank" rel="noopener">Johnson, Charles R.</a> (1991), [<em>Topics in Matrix Analysis</em>](<a href="https://books.google.com/?id=LeuNXB2bl5EC&amp;printsec=frontcover&amp;dq=isbn:9780521467131#v=onepage&amp;q=&quot;Kronecker" target="_blank" rel="noopener">https://books.google.com/?id=LeuNXB2bl5EC&amp;printsec=frontcover&amp;dq=isbn:9780521467131#v=onepage&amp;q=&quot;Kronecker</a> product”&amp;f=false), Cambridge University Press, <a href="https://www.wikiwand.com/en/International_Standard_Book_Number" target="_blank" rel="noopener">ISBN</a> <a href="https://www.wikiwand.com/en/Special:BookSources/978-0-521-46713-1" target="_blank" rel="noopener">978-0-521-46713-1</a>.</li>
<li>Jain, Anil K. (1989), <a href="https://books.google.com/?id=GANSAAAAMAAJ&dq=isbn%3A9780133361650&q=" target="_blank" rel="noopener"Kronecker+product""><em>Fundamentals of Digital Image Processing</em></a>, Prentice Hall, <a href="https://www.wikiwand.com/en/International_Standard_Book_Number" target="_blank" rel="noopener">ISBN</a> <a href="https://www.wikiwand.com/en/Special:BookSources/978-0-13-336165-0" target="_blank" rel="noopener">978-0-13-336165-0</a>.</li>
<li>Steeb, Willi-Hans (1997), <em>Matrix Calculus and Kronecker Product with Applications and C++ Programs</em>, World Scientific Publishing, <a href="https://www.wikiwand.com/en/International_Standard_Book_Number" target="_blank" rel="noopener">ISBN</a> <a href="https://www.wikiwand.com/en/Special:BookSources/978-981-02-3241-2" target="_blank" rel="noopener">978-981-02-3241-2</a></li>
<li>Steeb, Willi-Hans (2006), [<em>Problems and Solutions in Introductory and Advanced Matrix Calculus</em>](<a href="https://books.google.com/?id=CSDbVU1Eg3UC&amp;printsec=frontcover&amp;dq=isbn:9789812569165#v=onepage&amp;q=&quot;Kronecker" target="_blank" rel="noopener">https://books.google.com/?id=CSDbVU1Eg3UC&amp;printsec=frontcover&amp;dq=isbn:9789812569165#v=onepage&amp;q=&quot;Kronecker</a> product”&amp;f=false), World Scientific Publishing, <a href="https://www.wikiwand.com/en/International_Standard_Book_Number" target="_blank" rel="noopener">ISBN</a> <a href="https://www.wikiwand.com/en/Special:BookSources/978-981-256-916-5" target="_blank" rel="noopener">978-981-256-916-5</a></li>
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          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#定义"><span class="nav-number">1.</span> <span class="nav-text">定义</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#例子"><span class="nav-number">1.1.</span> <span class="nav-text">例子</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#特性"><span class="nav-number">2.</span> <span class="nav-text">特性</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#双线性和结合律"><span class="nav-number">2.1.</span> <span class="nav-text">双线性和结合律</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#混合乘积性质"><span class="nav-number">2.2.</span> <span class="nav-text">混合乘积性质</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#克罗内克和"><span class="nav-number">2.3.</span> <span class="nav-text">克罗内克和</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#谱"><span class="nav-number">2.4.</span> <span class="nav-text">谱</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#奇异值"><span class="nav-number">2.5.</span> <span class="nav-text">奇异值</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#与抽象张量积的关系"><span class="nav-number">2.6.</span> <span class="nav-text">与抽象张量积的关系</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#与图的乘积的关系"><span class="nav-number">2.7.</span> <span class="nav-text">与图的乘积的关系</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#转置"><span class="nav-number">2.8.</span> <span class="nav-text">转置</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#矩阵方程"><span class="nav-number">3.</span> <span class="nav-text">矩阵方程</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#参考文献"><span class="nav-number">4.</span> <span class="nav-text">参考文献</span></a></li></ol></div>
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